Theorem 1 ([1]})
Given a matrix \(A \in \mathbb {R}^{m \times n}\)  in matrix sample model, \(k \in \mathbb {N_{+}}\)  and \(\epsilon , \delta \in (0,1)\) , FKV(\(A,k,\epsilon ,\delta \) ) can output the approximate right singular matrix \(\widetilde{V} \in \mathbb {R}^{n \times k}\)  in \(O({\rm poly}(k, \frac{1}{\epsilon }, {\rm log}\,\frac{1}{\delta }))\)  samples and queries of \(A\)  with probability \(1-\delta \) , which satisfies
\(\Vert A \widetilde{V} \widetilde{V}^{T} - A\Vert _F^2 \le \min \limits _{\widetilde{A}:\,{\rm rank}(\widetilde{A})\le k} \Vert A - \widetilde{A} \Vert _F^2 + \epsilon \Vert A\Vert _F^2.\) 

